Finite Group Databasebeta

← #28611492

Subgroups — #28611492

Order 15104.

942 conjugacy classes of subgroups.

The subgroup lattice has 660 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837657C211yesmaximal, characteristic
2C2×#16837509C211yesmaximal, characteristic, complemented
3C2×#16837443C211yesmaximal, characteristic
4#21427128C211yesmaximal, characteristic
5#21427106C211yesmaximal, characteristic
6#21424837C211yesmaximal, characteristic, complemented
7#21424734C211yesmaximal, characteristic, complemented
8#21424727C211yesmaximal, characteristic, complemented
9#21424405C211yesmaximal, characteristic
10#21424400C211yesmaximal, characteristic
11#21424362C211yesmaximal, characteristic, complemented
12#21424348C211yesmaximal, characteristic, complemented
13C4×#14273659C211yesmaximal, characteristic, complemented
14#21423879C211yesmaximal, characteristic
15C59×#1969C211yesmaximal, characteristic, complemented
16C22×#11745659C2211yescharacteristic
17C2×#14273703C2211yescharacteristic
18C2×#14273703C2211yescharacteristic
19C2×Q8×Dic59C2211yescharacteristic
20C2×Q8×Dic59C2211yescharacteristic
21C2×#14273685C2211yescharacteristic
22C2×C4⋊C4×D59C2211yescharacteristic
23#16837657C418yescomplemented
24#16837509—24no—
25#16837443—24no—
26C2×#14273660C2211yescharacteristic
27C2×#14273659C2211yescharacteristic
28C2×#14273659C2211yescharacteristic
29C2×#14273658C2211yescharacteristic
30C2×#14273651C2211yescharacteristic
31C2×#14273651C2211yescharacteristic
32C2×#14273651C2211yescharacteristic
33C2×#14273651C2211yescharacteristic
34C2×(C2×D59)⋊C8C2211yescharacteristic
35C2×#14273637C2211yescharacteristic
36#16837299—21no—
37#16837298C2211yescharacteristic
38#16837298C2211yescharacteristic
39#16837294—21no—
40#16837291—21no—
41#16837289—21no—
42#16837284—21no—
43#16837276—21no—
44C4×C4⋊Dic59C2211yescharacteristic
45C4×C4⋊Dic59C2211yescharacteristic
46#16837253—21no—
47C4×(C2×D59)⋊C4C2211yescharacteristic
48#16837249—21no—
49#16837165C2211yescharacteristic
50#16837165C2211yescharacteristic
51#16837164C2211yescharacteristic
52#16837164C2211yescharacteristic
53#16837157C2211yescharacteristic
54#16837156C2211yescharacteristic
55C22×Q16×C59C2211yescharacteristic
56C2×#107×C59C2211yescharacteristic
57C2×#103×C59C2211yescharacteristic
58C2×#103×C59C2211yescharacteristic
59C2×C4×C8×C59C2211yescharacteristic
60C59×#391C2211yescharacteristic
61C59×#391C2211yescharacteristic
62C22×C4×D59—21no—
63C22×C4×D59C2311yescharacteristic
64C2×#11745659—22no—
65C2×#11745659—22no—
66C2×#11745659C2×C414yes—
67C22×C59⋊C8C2311yescharacteristic
68C22×C59⋊C8C2311yescharacteristic
69#14273703—22no—
70#14273703—22no—
71#14273703C2×C414yes—
72#14273703C2×C414yes—
73Q8×Dic59—22no—
74Q8×Dic59—22no—
75Q8×Dic59—41no—
76Q8×Dic59—41no—
77#14273685—22no—
78#14273685—41no—
79C2×C4⋊Dic59—21no—
80C2×C4⋊Dic59—21no—
81C2×C4⋊Dic59C2311yescharacteristic
82C2×C4⋊Dic59C2311yescharacteristic
83C2×C4⋊Dic59C2311yescharacteristic
84C2×C4⋊Dic59C2311yescharacteristic
85C2×Dic59⋊C4—21no—
86C2×Dic59⋊C4—21no—
87C2×Dic59⋊C4—21no—
88C2×Dic59⋊C4—21no—
89C2×Dic59⋊C4—21no—
90C4⋊C4×D59—22no—
91C4⋊C4×D59—41no—
92C2×(C2×D59)⋊C4—21no—
93C2×(C2×D59)⋊C4—21no—
94C2×(C2×D59)⋊C4—21no—
95C2×(C2×D59)⋊C4D411yescharacteristic
96C2×(C2×D59)⋊C4D411yescharacteristic
97C2×C4×Dic59—21no—
98C2×C4×Dic59—21no—
99C2×C4×Dic59—21no—
100C2×C4×Dic59—21no—
101C2×C4×Dic59—21no—
102C2×C4×Dic59C2311yescharacteristic
103#14273660—22no—
104#14273659C2×C414yescomplemented
105#14273659D414yes—
106#14273658—22no—
107#14273651—22no—
108#14273651—22no—
109#14273651C2×C414yescomplemented
110#14273651C2×C414yescomplemented
111(C2×D59)⋊C8—22no—
112#14273637—22no—
113C59⋊#95—21no—
114C59⋊#95—21no—
115C59⋊#95—21no—
116C59⋊#95—21no—
117C59⋊#95D411yescharacteristic
118C59⋊#95D411yescharacteristic
119C22×Q8×C59C2311yescharacteristic
120C22×Q8×C59C2311yescharacteristic
121C2×Q16×C59—22no—
122C2×Q16×C59—22no—
123C2×Q16×C59C2×C414yescomplemented
124C22×C8×C59C2311yescharacteristic
125C22×C8×C59C2311yescharacteristic
126C2×C4⋊C4×C59C2311yescharacteristic
127C2×C4⋊C4×C59C2311yescharacteristic
128C2×C42×C59C2311yescharacteristic
129#107×C59—22no—
130#103×C59—41no—
131#103×C59—41no—
132C4×C8×C59—22no—
133#95×C59—21no—
134#95×C59—21no—
135#95×C59—21no—
136#95×C59—21no—
137C23×D59C2×D411yescharacteristic
138C2×Q8×C59—21no—
139C2×Q8×C59—21no—
140C2×Q8×C59—21no—
141C2×Q8×C59—21no—
142C2×Q8×C59—21no—
143C2×Q8×C59—21no—
144C2×Q8×C59—21no—
145C2×Q8×C59—21no—
146C2×Q8×C59—21no—
147C2×Q8×C59—21no—
148C2×Q8×C59C22×C412yes—
149C2×Q8×C59C22×C412yes—
150C22×C4×C59—21no—
151C22×C4×C59—21no—
152C22×C4×C59—21no—
153C22×C4×C59—21no—
154C22×C4×C59C2×D411yescharacteristic
155C22×C4×C59C2×D411yescharacteristic
156C22×C4×C59C2411yescharacteristic
157C22×Dic59—21no—
158C22×Dic59—21no—
159C22×Dic59—21no—
160C22×Dic59—21no—
161C22×Dic59C2×D411yescharacteristic
162C22×Dic59C2×D411yescharacteristic
163C22×Dic59C2×D411yescharacteristic
164C2×C4×D59—21no—
165C2×C4×D59—21no—
166C2×C4×D59—21no—
167C2×C4×D59—21no—
168C2×C4×D59—42no—
169C2×C4×D59—42no—
170C2×C4×D59—81no—
171C2×C4×D59C22×C412yes—
172C2×C4×D59#4112yes—
173Q16×C59—28no—
174C2×C8×C59—21no—
175C2×C8×C59—21no—
176C2×C8×C59—21no—
177C2×C8×C59C22×C412yes—
178C2×C8×C59C2×D412yes—
179C2×C8×C59#4112yes—
180C4⋊C4×C59—21no—
181C4⋊C4×C59—21no—
182C4⋊C4×C59—21no—
183C4⋊C4×C59—21no—
184C42×C59—21no—
185C42×C59—21no—
186(C2×D59)⋊C4—24no—
187(C2×D59)⋊C4—24no—
188(C2×D59)⋊C4—22no—
189(C2×D59)⋊C4—41no—
190(C2×D59)⋊C4—42no—
191C4⋊Dic59—21no—
192C4⋊Dic59—21no—
193C4⋊Dic59—21no—
194C4⋊Dic59—21no—
195C4⋊Dic59—42no—
196C4⋊Dic59—42no—
197C4⋊Dic59C22×C412yes—
198C4⋊Dic59C22×C412yes—
199C4⋊Dic59C2×D412yes—
200C4⋊Dic59C2×D412yes—
201Dic59⋊C4—24no—
202Dic59⋊C4—24no—
203Dic59⋊C4—42no—
204Dic59⋊C4—42no—
205Dic59⋊C4—81no—
206C4×Dic59—21no—
207C4×Dic59—21no—
208C4×Dic59—42no—
209C4×Dic59—42no—
210C4×Dic59—42no—
211C4×Dic59—42no—
212C4×Dic59—81no—
213C2×C59⋊C8—21no—
214C2×C59⋊C8—21no—
215C2×C59⋊C8—21no—
216C2×C59⋊C8—21no—
217C2×C59⋊C8—21no—
218C2×C59⋊C8C22×C412yes—
219#11745659—28no—
220C23×C59C22×D411yescharacteristic
221C22×D59—21no—
222C22×D59—21no—
223C22×D59—21no—
224C22×D59—21no—
225C22×D59—21no—
226C22×D59—21no—
227C22×D59C4×D412yes—
228Q8×C59—22no—
229Q8×C59—22no—
230Q8×C59—22no—
231Q8×C59—22no—
232Q8×C59—41no—
233Q8×C59—41no—
234Q8×C59—41no—
235Q8×C59—41no—
236C2×C4×C59—21no—
237C2×C4×C59—21no—
238C2×C4×C59—22no—
239C2×C4×C59—22no—
240C2×C4×C59—21no—
241C2×C4×C59—21no—
242C2×C4×C59—21no—
243C2×C4×C59—41no—
244C2×C4×C59—41no—
245C2×C4×C59—41no—
246C2×C4×C59—41no—
247C2×C4×C59—41no—
248C2×C4×C59—41no—
249C2×C4×C59—41no—
250C2×C4×C59—41no—
251C2×C4×C59—41no—
252C2×C4×C59—41no—
253C2×C4×C59#13712yes—
254C2×C4×C59C23×C411yescharacteristic
255C2×C4×C59C22×D411yescharacteristic
256C2×C4×C59C2×#4111yescharacteristic
257C2×C4×C59C2×#4111yescharacteristic
258C2×C4×C59H(5,𝔽2)11yescharacteristic
259C2×C4×C592−1+411yescharacteristic
260C2×Dic59—21no—
261C2×Dic59—21no—
262C2×Dic59—22no—
263C2×Dic59—22no—
264C2×Dic59—21no—
265C2×Dic59—21no—
266C2×Dic59—22no—
267C2×Dic59—22no—
268C2×Dic59—21no—
269C2×Dic59—21no—
270C2×Dic59—21no—
271C2×Dic59—21no—
272C2×Dic59—41no—
273C2×Dic59—41no—
274C2×Dic59—41no—
275C2×Dic59—41no—
276C2×Dic59—41no—
277C2×Dic59—41no—
278C2×Dic59—41no—
279C2×Dic59—41no—
280C2×Dic59C4×D412yes—
281C4×D59—22no—
282C4×D59—22no—
283C4×D59—22no—
284C4×D59—22no—
285C4×D59—84no—
286C8×C59—41no—
287C8×C59C4×D414yes—
288C59⋊C8—22no—
289C59⋊C8—41no—
290#23038—591nomaximal, Sylow
291C22×C59C2×C4×D411yescharacteristic
292C22×C59D4211yescharacteristic
293C22×C59#54711yescharacteristic
294C22×C59C2×#13711yescharacteristic
295C22×C59#57511yescharacteristic
296C22×C59#57711yescharacteristic
297C22×C59#57811yescharacteristic
298C2×D59—22no—
299C2×D59—22no—
300C2×D59—22no—
301C2×D59—41no—
302C2×D59—41no—
303C2×D59—41no—
304C2×D59—41no—
305C4×C59—22no—
306C4×C59—41no—
307C4×C59—42no—
308C4×C59—42no—
309C4×C59—81no—
310C4×C59—81no—
311C4×C59C2×C4×D411yescharacteristic
312C4×C59#52011yescharacteristic
313C4×C59#54811yescharacteristic
314C4×C59#55311yescharacteristic
315Dic59—22no—
316Dic59—42no—
317Dic59—42no—
318Dic59—42no—
319Dic59—41no—
320Dic59—41no—
321Dic59—42no—
322C2×#470—591no—
323C2×#441—591no—
324C2×#428—591no—
325#2362—591no—
326#2360—591no—
327#1969D5911yescharacteristic, complemented
328#1944—591no—
329#1928—591no—
330#1927—591no—
331#1873—591no—
332#1872—591no—
333#1865—591no—
334#1863—591no—
335C4×#106—591no—
336#1753—591no—
337C2×C59#236111yescharacteristic
338C2×C59C4×#13711yescharacteristic
339C2×C59#294711yescharacteristic
340C2×C59#327511yescharacteristic
341C2×C59#327611yescharacteristic
342C2×C59#334411yescharacteristic
343C2×C59#334911yescharacteristic
344D59—42no—
345C22×Q16C2×D5911yescharacteristic
346C22×M16—591no—
347C2×C22⋊Q8—591no—
348C2×C22⋊Q8—591no—
349C2×C4×Q8—591no—
350C2×C4×Q8—591no—
351C2×#117—591no—
352C22×C4⋊C4—591no—
353#470—598no—
354#441—1184no—
355#428—1184no—
356C2×#107—591no—
357C2×#107C2×D5911yescharacteristic
358C2×#106—591no—
359C2×#106—591no—
360C2×#106—591no—
361C2×#103—591no—
362C2×#103—591no—
363C2×#103—591no—
364C2×#103—591no—
365C2×#103C2×D5911yescharacteristic
366C2×#103C2×D5911yescharacteristic
367C2×C22⋊C8—591no—
368C2×#97—591no—
369C2×C4×C8C2×D5911yescharacteristic
370#391C2×D5911yescharacteristic
371#391C2×D5911yescharacteristic
372#389—591no—
373#389—591no—
374#389—1181no—
375#388—1181no—
376#388—1181no—
377#387—1181no—
378#387—1181no—
379#385—1181no—
380C4×C4⋊C4—591no—
381C4×C4⋊C4—591no—
382C4×C4⋊C4—1181no—
383C4×C22⋊C4—591no—
384C4×C22⋊C4—1181no—
385#340—591no—
386#340—591no—
387#340—591no—
388#340—591no—
389#336—591no—
390#336—591no—
391C59#2303811yescharacteristic, complemented, Sylow
392C22×Q8C22×D5911yescharacteristic
393C22×Q8C22×D5911yescharacteristic
394C23×C4—591no—
395C23×C4—1181no—
396C2×Q16—22no—
397C2×Q16—22no—
398C2×Q16C4×D5914yescomplemented
399C2×M16—594no—
400C2×M16—1182no—
401C2×M16—1182no—
402C22×C8—591no—
403C22×C8—591no—
404C22×C8C22×D5911yescharacteristic
405C22×C8C22×D5911yescharacteristic
406C22⋊Q8—594no—
407C22⋊Q8—594no—
408C22⋊Q8—1182no—
409C22⋊Q8—1182no—
410C4×Q8—1182no—
411C4×Q8—1182no—
412C4×Q8—2361no—
413C4×Q8—2361no—
414#117—1182no—
415#117—2361no—
416C2×C4⋊C4—591no—
417C2×C4⋊C4—591no—
418C2×C4⋊C4—591no—
419C2×C4⋊C4—591no—
420C2×C4⋊C4—1182no—
421C2×C4⋊C4—1181no—
422C2×C4⋊C4—1181no—
423C2×C4⋊C4—1181no—
424C2×C4⋊C4—1181no—
425C2×C4⋊C4—1181no—
426C2×C4⋊C4—1181no—
427C2×C4⋊C4—1181no—
428C2×C4⋊C4—2361no—
429C2×C4⋊C4C22×D5911yescharacteristic
430C2×C4⋊C4C22×D5911yescharacteristic
431C2×C22⋊C4—591no—
432C2×C22⋊C4—591no—
433C2×C22⋊C4—1181no—
434C2×C22⋊C4—1181no—
435C2×C22⋊C4—1181no—
436C2×C42—591no—
437C2×C42—1181no—
438C2×C42—1181no—
439C2×C42—1181no—
440C2×C42—1181no—
441C2×C42—1181no—
442C2×C42C22×D5911yescharacteristic
443#107—22no—
444#107—1182no—
445#106—594no—
446#106—594no—
447#106—1182no—
448#103—41no—
449#103—41no—
450#103—594no—
451#103—594no—
452#103—1182no—
453#103—1182no—
454C22⋊C8—1182no—
455#97—1182no—
456C4×C8—22no—
457#95—21no—
458#95—21no—
459#95—21no—
460#95—21no—
461#95—591no—
462#95—591no—
463#95—1181no—
464#95—1181no—
465#95—1181no—
466#95—1181no—
467C24—591no—
468C2×Q8—21no—
469C2×Q8—21no—
470C2×Q8—21no—
471C2×Q8—21no—
472C2×Q8—21no—
473C2×Q8—21no—
474C2×Q8—21no—
475C2×Q8—21no—
476C2×Q8—21no—
477C2×Q8—21no—
478C2×Q8C2×C4×D5912yes—
479C2×Q8C2×C4×D5912yes—
480C22×C4—21no—
481C22×C4—21no—
482C22×C4—21no—
483C22×C4—21no—
484C22×C4—591no—
485C22×C4—592no—
486C22×C4—592no—
487C22×C4—591no—
488C22×C4—591no—
489C22×C4—1181no—
490C22×C4—1181no—
491C22×C4—1181no—
492C22×C4—1181no—
493C22×C4—1181no—
494C22×C4—1181no—
495C22×C4—1181no—
496C22×C4—1181no—
497C22×C4—4721no—
498C22×C4—2362no—
499C22×C4—2362no—
500C22×C4D4×D5911yescharacteristic
501C22×C4D4×D5911yescharacteristic
502C22×C4C23×D5911yescharacteristic
503Q16—28no—
504M16—1188no—
505C2×C8—21no—
506C2×C8—21no—
507C2×C8—21no—
508C2×C8—592no—
509C2×C8—1181no—
510C2×C8—1181no—
511C2×C8—1181no—
512C2×C8—1181no—
513C2×C8—1181no—
514C2×C8C2×C4×D5912yes—
515C2×C8D4×D5912yes—
516C2×C8#1174568712yes—
517C4⋊C4—21no—
518C4⋊C4—21no—
519C4⋊C4—21no—
520C4⋊C4—21no—
521C4⋊C4—592no—
522C4⋊C4—592no—
523C4⋊C4—592no—
524C4⋊C4—592no—
525C4⋊C4—1184no—
526C4⋊C4—1184no—
527C4⋊C4—1181no—
528C4⋊C4—1181no—
529C4⋊C4—1181no—
530C4⋊C4—1181no—
531C4⋊C4—4721no—
532C4⋊C4—2362no—
533C4⋊C4—2362no—
534C4⋊C4—2362no—
535C4⋊C4—2362no—
536C22⋊C4—1184no—
537C22⋊C4—1184no—
538C22⋊C4—1182no—
539C22⋊C4—2361no—
540C22⋊C4—2362no—
541C42—21no—
542C42—21no—
543C42—1181no—
544C42—1181no—
545C42—4721no—
546C42—2362no—
547C42—2362no—
548C42—2362no—
549C42—2362no—
550C23—592no—
551C23—1181no—
552C23—1181no—
553C23—1181no—
554C23—1181no—
555C23—1181no—
556C23—1181no—
557C23C2×D4×D5911yescharacteristic
558Q8—22no—
559Q8—22no—
560Q8—22no—
561Q8—22no—
562Q8—41no—
563Q8—41no—
564Q8—41no—
565Q8—41no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—592no—
584C2×C4—1181no—
585C2×C4—1181no—
586C2×C4—1182no—
587C2×C4—1182no—
588C2×C4—1182no—
589C2×C4—1182no—
590C2×C4—1181no—
591C2×C4—1181no—
592C2×C4—1181no—
593C2×C4—1181no—
594C2×C4—1181no—
595C2×C4—1181no—
596C2×C4—1182no—
597C2×C4—1182no—
598C2×C4—1182no—
599C2×C4—1182no—
600C2×C4—4724no—
601C2×C4—2361no—
602C2×C4—2361no—
603C2×C4—2361no—
604C2×C4—2361no—
605C2×C4—2361no—
606C2×C4—2361no—
607C2×C4—2361no—
608C2×C4—2361no—
609C2×C4#1427375912yes—
610C2×C4C22×C4×D5911yescharacteristic
611C2×C4C2×D4×D5911yescharacteristic
612C2×C4#41×D5911yescharacteristic
613C2×C4C2×#1174568711yescharacteristic
614C2×C4#1427377211yescharacteristic
615C2×C4#1427377411yescharacteristic
616C8—41no—
617C8—1182no—
618C8—2361no—
619C8(C22⋊D59)⋊C414yes—
620C22—1182no—
621C22—1182no—
622C22—1182no—
623C22—2361no—
624C22—2361no—
625C22—2361no—
626C22—2361no—
627C22C4×D4×D5911yescharacteristic
628C22#1683803711yescharacteristic
629C22#1683805211yescharacteristic
630C22C2×#1427375911yescharacteristic
631C22#1683819911yescharacteristic
632C22#1683820911yescharacteristic
633C22#1683821511yescharacteristic
634C4—22no—
635C4—42no—
636C4—42no—
637C4—41no—
638C4—81no—
639C4—81no—
640C4—1182no—
641C4—2361no—
642C4—2362no—
643C4—2362no—
644C4—2361no—
645C4—2362no—
646C4—2362no—
647C4C2×(C22⋊D59)⋊C411yescharacteristic
648C4#1683788111yescharacteristic
649C4#1683806011yescharacteristic
650C4#1683809911yescharacteristic
651C2—2362no—
652C2#2142711711yescharacteristic
653C2C4×#1427375911yescharacteristic
654C2#2143293411yescharacteristic
655C2#2143572611yescharacteristic
656C2#2143574111yescharacteristic
657C2#2143655711yescharacteristic
658C2#2143663211yescharacteristic
subgroupG/H
centerC23C2×D4×D59
commutatorC2×C4×C59C23×C4
FrattiniC22×C4C23×D59
FittingC59×#1969C2
socleC23×C59C22×D4
prime radicalC23×D59C2×D4
cosocleC22×C4×C59C24
hypercenter#1969D59
hypocenterC59#23038
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1969

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C2×#16837509C2semidirect
6. #21424837C2semidirect
7. #21424734C2semidirect
8. #21424727C2semidirect
11. #21424362C2semidirect
12. #21424348C2semidirect
13. C4×#14273659C2semidirect
15. C59×#1969C2semidirect
23. #16837657C4semidirect
104. #14273659C2×C4semidirect
109. #14273651C2×C4semidirect
110. #14273651C2×C4semidirect
123. C2×Q16×C59C2×C4semidirect
327. #1969D59semidirect
391. C59#23038semidirect
398. C2×Q16C4×D59semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#168376572
C2×#168375092
C2×#168374432
#214271282
#214271062
#214248372
#214247342
#214247272
#214244052
#214244002
#214243622
#214243482
C4×#142736592
#214238792
C59×#19692
#2303859